After dealing with various techniques to evaluate limits we now provide proofs of the results on which these techniques are founded. This material is not difficult but definitely somewhat abstract and may not be suitable for beginners who are more interested in learning techniques and solving limit problems. But those who are interested in the justification of these techniques must pay great attention to what follows.

Proofs of Rules of Limits

We provide proofs for some of the rules and let the reader provide proofs for remaining rules based on similar line of argument. First we start with rule dealing with inequalities:If $f(x) \leq g(x)$ in the neighborhood of $a$ then $\lim_{x \to a}f(x) \leq \lim_{x \to a}g(x)$ provided both these limits exist.

In lasttwo posts we have developed basic concepts and rules of limits. Continuing our journey further we now introduce certain powerful tools which help us in evaluation of limits of complicated expressions. We start with the simplest technique first.

Limits using Logarithms

In case we need to evaluate the limit of an expression of type $\{f(x)\}^{g(x)}$ then we can take logarithm and then the evaluation of limits becomes simpler. We will first illustrate the technique through an example and then provide the justification.

After the definitions and basic examples in Part 1, we now focus on the rules of evaluation of limits which will be highly useful in solving various limit problems. We will postpone the proofs of these rules to the last post in the series to avoid any distraction.

Rules of Limits

In the following rules we assume that the functions described are defined in a certain neighborhood of $a$ except possibly at point $a$. All the relations between the functions (if any) also hold in this neighborhood of point $a$ (except possibly at point $a$).

Introduction

While looking at certain limit problems posed in math.stackexchange.com (henceforth to be called MSE) I found that most beginners studying limits are living in a fantasy world consisting of vague notions, infinities and what not. I too had my share of such experiences during my time as a student learning calculus but I was lucky enough to get over with this phase very quickly through the help of "A Course of Pure Mathematics".

Regarding the answers posted on MSE I found that most of the answers although correct were not suitable for beginners studying limits. Some answers suggested that their authors themselves had the same vague notions but they somehow managed to avoid their pitfalls. Some other answers were using sophisticated techniques which involved deeper concepts than the concept of limits itself. And there were some heated arguments favoring one approach over another.

Therefore I decided to write a series of posts providing a step by step approach to solve limit problems encountered in an introductory calculus course. I have tried to split the whole topic into $4$ posts and I believe that the gist of each post can be assimilated in not more than $2$ hours and that's the logic behind the title of this series.